Windows 7 USB DVD Download Tool fails on UEFI-boot systems

If your system is set to EFI/UEFI boot, and you try to use the Windows 7 USB/DVD download tool to make a Windows 7 bootable USB stick, it will fail with the following error:

Files copied successfully. However, we were unable to run bootsect to make the USB device bootable. If you need assistance with bootsect, please click the “Online Help” link above for more information.

The reason? Well, I found this post that had instructions on running bootsect.exe manually. I did that, I received this message:

This tool can only be run on systems booted using a PC/AT BIOS. This system was booted using EFI or some other firmware type.